Struct ArrudaBoyce

pub struct ArrudaBoyce {
    pub bulk_modulus: Scalar,
    pub shear_modulus: Scalar,
    pub number_of_links: Scalar,
}

The Arruda-Boyce hyperelastic constitutive model.¹

Parameters

• The bulk modulus $\kappa$.
• The shear modulus $\mu$.
• The number of links $N_b$.

External variables

• The deformation gradient $\mathbf{F}$.

Internal variables

• None.

Notes

• The nondimensional end-to-end length per link of the chains is $\gamma=\sqrt{\mathrm{tr}(\mathbf{B}^*)/3N_b}$.
• The nondimensional force is given by the inverse Langevin function as $\eta=\mathcal{L}^{-1}(\gamma)$.
• The initial values are given by $\gamma_0=\sqrt{1/3N_b}$ and $\eta_0=\mathcal{L}^{-1}(\gamma_0)$.
• The Arruda-Boyce model reduces to the Neo-Hookean model when $N_b\to\infty$.

---

1. E.M. Arruda and M.C. Boyce, J. Mech. Phys. Solids 41, 389 (1993). ↩

Fields

bulk_modulus: Scalar

The bulk modulus $\kappa$.

shear_modulus: Scalar

The shear modulus $\mu$.

number_of_links: Scalar

The number of links $N_b$.

Implementations

impl ArrudaBoyce

pub fn number_of_links(&self) -> &Scalar

Returns the number of links.

Trait Implementations

impl Debug for ArrudaBoyce

fn fmt(&self, f: &mut Formatter<'_>) -> Result

Formats the value using the given formatter.

impl Elastic for ArrudaBoyce

fn cauchy_stress( &self, deformation_gradient: &DeformationGradient, ) -> Result<CauchyStress, ConstitutiveError>

\boldsymbol{\sigma}(\mathbf{F}) = \frac{\mu\gamma_0\eta}{J\gamma\eta_0}\,{\mathbf{B}^*}' + \frac{\kappa}{2}\left(J - \frac{1}{J}\right)\mathbf{1}

fn cauchy_tangent_stiffness( &self, deformation_gradient: &DeformationGradient, ) -> Result<CauchyTangentStiffness, ConstitutiveError>

\begin{aligned}
\mathcal{T}_{ijkL}(\mathbf{F}) =
\,& \frac{\mu\gamma_0\eta}{J^{5/3}\gamma\eta_0}\left(\delta_{ik}F_{jL} + \delta_{jk}F_{iL} - \frac{2}{3}\,\delta_{ij}F_{kL}- \frac{5}{3} \, B_{ij}'F_{kL}^{-T} \right)
\\
&+ \frac{\mu\gamma_0\eta}{3J^{7/3}N_b\gamma^2\eta_0}\left(\frac{1}{\eta\mathcal{L}'(\eta)} - \frac{1}{\gamma}\right)B_{ij}'B_{km}'F_{mL}^{-T} + \frac{\kappa}{2} \left(J + \frac{1}{J}\right)\delta_{ij}F_{kL}^{-T}
\end{aligned}

fn first_piola_kirchhoff_stress( &self, deformation_gradient: &DeformationGradient, ) -> Result<FirstPiolaKirchhoffStress, ConstitutiveError>

Calculates and returns the first Piola-Kirchhoff stress.

fn first_piola_kirchhoff_tangent_stiffness( &self, deformation_gradient: &DeformationGradient, ) -> Result<FirstPiolaKirchhoffTangentStiffness, ConstitutiveError>

Calculates and returns the tangent stiffness associated with the first Piola-Kirchhoff stress.

fn second_piola_kirchhoff_stress( &self, deformation_gradient: &DeformationGradient, ) -> Result<SecondPiolaKirchhoffStress, ConstitutiveError>

Calculates and returns the second Piola-Kirchhoff stress.

fn second_piola_kirchhoff_tangent_stiffness( &self, deformation_gradient: &DeformationGradient, ) -> Result<SecondPiolaKirchhoffTangentStiffness, ConstitutiveError>

Calculates and returns the tangent stiffness associated with the second Piola-Kirchhoff stress.

impl Hyperelastic for ArrudaBoyce

fn helmholtz_free_energy_density( &self, deformation_gradient: &DeformationGradient, ) -> Result<Scalar, ConstitutiveError>

a(\mathbf{F}) = \frac{3\mu N_b\gamma_0}{\eta_0}\left[\gamma\eta - \gamma_0\eta_0 - \ln\left(\frac{\eta_0\sinh\eta}{\eta\sinh\eta_0}\right) \right] + \frac{\kappa}{2}\left[\frac{1}{2}\left(J^2 - 1\right) - \ln J\right]

impl Solid for ArrudaBoyce

fn bulk_modulus(&self) -> &Scalar

Returns the bulk modulus.

fn shear_modulus(&self) -> &Scalar

Returns the shear modulus.